Proposition 21
 

A regular Polygon of an even number of sides being inscribed in a circle as ABC...A'...C'B'A' so that AA' is a diameter if two angular points next but one to each other as B, B' be joined, and the other lines parallel to BB' and joining pairs of angular points be drawn as CC', DD',...., then

(BB' + CC' +.... ) / AA' = A'B / BA
 
The two angle in red color are equal due to parallel lines. The two blue lines are parallel and the black line crosses the two blue parallel line. therefore the two red angles are equal and we know that the black line between two parallel line is perpendicular to diameter of the great circle.